PRECISE MEASUREMENT Precise measurement

PRECISE MEASUREMENT BY NON-LINEAR SENSOR WITH STAIRCASE REFERENCE SIGNAL

Sergey Sibeykin

Abstract. We use a staircase reference signal for measurement of electrical and non-electrical values by a nonlinear sensor. A sensor with a frequency output signal is preferable. We can measure the input signal itself (e.g. acceleration) or integral from it (e.g. velocity). Signals are considered as random variables. The result was found that the error of measurement is minimum if reference signal has the same duration and raw moments as the input signal. The reference signal can cover only a part of the input signal. A concept of negative probability was introduced to describe the composition of signal and the method of its transformation. The physical interpretation of the positive and negative probabilities is subsequent direct or inverse integration of signal.

Index Terms: Measurement; Measurement errors; Acceletation measurement; Velocity measurement; Voltmeters; Probability.

I. Introduction This method of measurement can be applied in various fields (in trains, missiles, aviation, automobiles, submarines, and robotics) to measure acceleration, velocity, vibrations, direct/alternating voltage, etc.
A task is to measure physical values by a n
onlinear sensor with minimal error. The output signal of sensor is a frequency which permits to calculate parameters of signal, including moments. This sensor may be piezo-element, since its frequency depends on the pressure of inertial mass. Measured values may be: pressure, acceleration, vibration, etc. We can measure also direct or alternating voltage that applies to varicap. The piezo-element (or varicap) is part of the oscillation circuit. A measured value may also be an integral from the input signal (velocity, if the input signal is acceleration).

Let's consider the measurement of velocity (Fig. 1). Our input signal is acceleration x(t) that is measured by a piezo-accelerometer. Counter of impulses measures its output signal (frequency). The code of Counter is proportional the reading W(t) of a measured velocity V(t).

In the simplest case we find the measured velocity V(t) that has got some value V_r. This task may be solved as follows:

We put the reading W_r of the value V_r in Register by integration of a constant reference signal. This acceleration is due to Earth’s gravity g = 9.81m/sec². Therefore, time of integration Tr = V_r/g. In the process of measuring, the Digital Comparator compares the codes of Counter and Register. When W (t) = W_r, Comparator gives the signal that the velocity V(t) has a value V_r. Lets denote this moment of time by T. However, since the sensor may not be a perfect machine, the real velocity V(T) may be different from V_r by a value of DV. Error of measurement DV = V(T) - V_r .

We can decrease the error DV by replacing the constant reference signal g by the staircase reference signal X_r, which approximates the input signal x(t) (Fig. 2a, here the input signal x(t) is linear). Then we have the mutual compensation of errors for W(T) from the input signal x(t) and Wr from the reference signal Xr [1]. Therefore, when W(T) = W_r, V(T) » V_r, the error DV is at a minimum. The staircase reference signal is preferable because it is easier to provide parameters of stairs X_i and T_i with higher accuracy.

Usually, we know approximately an acceleration that we will measure. Therefore, we know the expected velocity V_e at a time T, and can assume that V_r = V_e. We receive the values X_i from -g to +g by rotation of accelerometer relative to the local vertical line. We integrate X_i for a time T_i in order to obtain V_r = X₁×T₁ + X₂×T₂ + X₃×T₃+..., and put the reading W_r of a reference velocity V_r in Register.

Our task is to find the optimal reference signals, i.e. what values Xi and Ti minimize the error of measurement DV.

II. Assumptions
a) We will consider the input x(t) and the reference Xr signals as random variables. The main characteristic of random variable X is probability function p(x). It has two properties [2]:

p(x) is non-negative: p(x) ³ 0		(1)
	¥
integral of p(x) is equal to 1:	ò p(x)×dx = 1	(2)
	-¥

For the linear input signal x(t), depicted on Fig. 2, probability function p(x) has an even distribution. Then we can derive from (2) p(x) = 1/Xmax.
The probability function p(X_r) of staircase reference signal with stairs is a composition of functions d_i, which is proportional to the delta-function d [3]:

	_n
p(X_r) =	å d _i	(3)
	^{i = 1}

here d_i = (T_i/T)×d.

Probability functions p(x) and p(X_r) are depicted on Fig. 2b.

We will also use moments for describing a random variable X. The raw moment M_i of i-th order is:

	¥
M_i =	ò p(x)×Xⁱ×dx	(4)
	-¥

We can also define the moments Mi as an expectation for variable x(t) of degree i:

	_T
M_i = (1/T)	òx(t)ⁱ×dt	(5)
	⁰

b) We describe the transfer function of the sensor as a Maclaurin series:

y = a₀ + (1+a₁)×x + a₂×x² + a₃×x³ +... = x + a₀ + a₁×x + a₂×x + a₃×x + ...

(6)

Here a₀ is the bias, a₁ is the error of scale factor, and a₂, a₃ ... describe the nonlinear error.
An ideal sensor has the transfer function y = x.
c) We suppose the parameters of the sensor have short term stability, i.e. values a₀, a₁, a₂ ... do not change from calibration till measurement.
d) Sensor does not have hysteresis.

III. Algorithm for an optimal reference signal
Let’s try to minimize the error of measurement. To do this, we need to find and equate the reading of velocity W(t) at a time T and a value W_r.
Taking into account (6), the reading W(T) is:

	_T	_T	_T	_T	_T	_T
W(T) =	ò y(t)dt =	ò x(t)dt =	ò a₀dt =	ò a₁×x(t)dt =	ò a₂×x²(t)dt =	ò a₃×x³(t)dt + ...	(7)
	⁰	⁰	⁰	⁰	⁰	⁰

Using (5), following is what we get from (7):

W_r = T×M₁ + a₀×T + a₁×T×M₁ +T×M₁ + a₂×T×M₂ + a₃×T×M₃ + ...

(8)

The reading of an integral from the reference signal X_r, i.e. the reading of a reference velocity V_r:

	t₁	t₂	t₃
W_r =	òY₁dt +	òY₂dt +	òY₃dt + ...
	⁰	⁰	⁰

The values Y_i of the staircase reference signal X_r are constant, therefore:

W_r = T₁×Y₁ + T₂×Y₂ + T₃×Y₃ + ...

(9)

Taking into account (6), we obtain:

W_r = T₁×(X₁+a₀+ a₁×X₁+ a₂×X₁²+ a₃×X₁³+...) + T₂×(X₂+a₀+ a₁×X₂+ a₂×X₂²+ a₃×X₂³+...) + T₃×(X₃+a₀+ a₁×X₃+ a₂×X₃²+ a₃×X₃³+...) + ...
or
W_r = T₁X₁+T₂X₂+T₃X₃+... + a₀×(T₁+T₂+T₃+...) + a₁× (T₁×X₁ +T₂×X₂ +T₃×X₃+...) + a₂× (T₁×X₁² +T₂×X₂² +T₃×X₃²+...) + a₃× (T₁×X₁³ +T₂×X₂³ +T₃×X₃³+...) + ...		(10)

Equating (8) and (10), put the terms with a_i to the right side of the equation:
T×M₁ - (T₁×M₁+ T₂×M₂+ T₃×M₃+...) = a₀×(T₁+T₂+T₃ - T) + a₁×(T₁×X₁2+T₂×X₂+T₃×X₃ + ... - T×M₁) + a₂×(T₁×X₁²+T₂×X₂²+T₃×X₃² + ... - T×M₂) + a₃×(T₁×X₁³+T₂×X₂³+T₃×X₃³ + ... - T×M₃) + ...
Here, value T×M1 is the velocity V(T), and the value T₁×X₁+ T₂×X₂+ T₃×X₃+... is V_r.

Therefore, the left part of this equation is the error of measurement DV=T×M₁ -(T₁×X₁ + T₂×X₂ + T₃×X₃ +...) = V(T) - V_r.
Then

DV = a₀×(T₁+T₂+T₃ - T) + a₁×(T₁×X₁+T₂×X₂+T₃×X₃ - T×M₁) + a₂×(T₁×X₁²+T₂×X₂²+T₃×X₃² - T×M₂) + a₃×(T₁×X₁³+T₂×X₂³+T₃×X₃³ - T×M₃) + ...

(11)

Error DV is equal to zero if the terms in parentheses on the right part of (11) are zero. Let us denote the terms of order >n as insignificant, i.e a_i = 0 if i > n. Then conditions for DV = 0 are:

T	= T₁	+ T₂	+ T₃ +...	+ T_n ;	(12)
T×M₁	= T₁×X₁	+ T₂×X₂	+ T₃×X₁ +...	+ T_n×X_n ;
T×M₂	= T₁×X₁²	+ T₂×X₂²	+ T₃×X₃² +...	+ T_n×X_n² ;
T×M₃	= T₁×X₁³	+ T₂×X₂³	+ T₃×X₃³ +...	+ T_n×X_n³ ;
. . . . . . . . . . .
T×M_n	= T₁×X₁ⁿ	+ T₂×X₂ⁿ	+ T₃×X₃ⁿ +...	+ T_n×X_nⁿ .

Let us multiply and divide the right parts of (12) by the durations of reference signal Tr = T₁ + T₂ + T₃ +...+ T_n. We obtain in these right parts the products of T_r and moments M_r,i of the staircase reference signal X_r. We can write (12) in short form as:

T×M₁ = T_r×M_r,i, i = 0, 1, 2, ... n-1,

(13)

here is the quantity of stairs of the reference signal X_r.

Therefore, the error of measurement is minimized if the durations and the moments of input and reference signals are equal.

The staircase reference signal with stairs can decrease the error DV from the first terms of transfer function (6) to zero, if we precisely evaluate the moments M_i of input signal.

To find the optimal reference signal, we have to decide on the task of moments’ problem, i.e. find the parameters of the reference signal on its moments, which are equal to moments of input signal.

For example, let the reference signal X_r has four stairs (n = 4), and the input signal x(t) be linear. In accordance with (4), or (5), the first, second, and third raw moments of this signal are:
M₁ = X_max/2, M₂ = X²_max/3, M₃ = X³_max/4.

Let the values X_i of the staircase reference signal X_r be uniformly distributed along the measurement's scale: X₁ = 0; X₂ = X_max/3; X₃ = 2×X_max/3; X₄ = X_max.

We know the values M_i and X_i, and therefore we can find the values T_i by solving a system of equations (12) via determinants. Then T₁ = T/8; T₂ = 3×T/8; T₃ = 3×T/8; T₄ = T/8.

IV. Error of measurement

	________________
The resulting relative error of measurement is d_meas = Ö	d²_meth + d²_res + d²_r .

d_meth is a methodical error, because it is the error of evaluation for moments M_i of input signal x(t).

d_meth » d_d × d_e

(14)

Here:
d_d is a relative error of direct measurement (without application of reference signals);
d_e is a relative error of evaluation of the first raw moment M₁.
The proof for (14) is omitted.
d_res is a residual error because of (n+1)-th, (n+2)-th, ...order terms of transfer function (6).
d_r is an error of the reading W_r of the reference velocity V_r.

V. Case when reference signal covers part of input signal
Our reference signal is limited by a value ±g, but we often measure much more acceleration. The new method works properly in this case as well.

Let acceleration changes itself in the range from 0 to 10g linearly, Xmax=10g. Let the values
X₁ = -g; X₂ = -0.5g; X₃ = 0.5g; X₄ = g.

Then by solving (12) we obtain the next durations of stairs:
T₁ = -143.7778×T, T₂ = 305.1111×T, T₃ = -348.2222×T, T₄ = 187.8889×T.

Values T₁ and T₃ are negative. This means Counter (Integrator) must subtract impulses for times T₁ and T₃ (Inverse integration).
We see that the reference signals may cover only a part of the input signal, or otherwise go out of its range. Then in accordance with (9), the reading W_r of reference velocity V_r is the difference of great numbers Y_i×T_i. This limits the accuracy of W_r and, therefore, the accuracy of measurement. The range of the reference signal may be only in several times less than the range of the input signal.

VI. A continuous measurement
We can introduce various values Wr at many points of the trajectory. Therefore, the measurement of velocity V(t) becomes continuous. We have the next sequence of operations:
6.1. Calibration. We find the readings Y₁, Y₂, Y₃,…Y_n for chosen values X₁, X₂, X₃,…X_n of the reference signal and save them in Register.
6.2. Measurement.
a) Evaluation of moments M₁, M₂, M₃, ... M_n-1 for the next point of measurement. For this we can find the expected velocity V_e by extrapolation from measured values V(t).
b) Calculation of the values T_1, T₂, T₃, ... T_n by solving (12) and the expected reading W_r of the reference velocity V_r according to
(9):
W_r = T₁×Y₁ + T₂×Y₂ + T₃×Y₃ + ... + T_n×Y_n.
Here, we can calculate W_r by multiplication and addition (subtraction, if T_i is negative) instead of integration. This significantly decreases the time of preparation for measurement (i.e. the time between the next and previous points of measurement is very short).
c) comparing W(t) and W_r. When W(t)=W_r, Comparator gives a signal that V(t) has got V_r.
If we find moments M₁, M₂, M₃, ... M_n-1 in the process of measurement, the difference between expected values of these moments and their real values is minimized. According to (14), accuracy of measurement is high. A computer calculation showed that the relative error of measurement may get 10^-5 if nonlinear error of sensor is a few percentage points.

VII. Measurement of voltage
We can measure the average value of input signal x(t), for example, voltage, too. The measured voltages apply to a varicap , which is part of the LC circuit. Voltage converts to a frequency:

1
f = ¾¾¾
¾
2p ÖLC

The nonlinear error is minimized by this method. The staircase reference signal X_r is created by connecting the stable sources of voltage X₁, X₂, ... X_n for times T₁, T₂, ... T_n to varicap. In comparison with the voltmeter of dual slope integration, this voltmeter has a low level of flicker noise (1/f noise). The resolution of this voltmeter can get a few microvolts by a reference signal with four stairs.

VIII. One Mathematical Consequence
The proposed method has one interesting consequence.
Let us find the probability functions p(Xr) of the staircase reference signal X_r covering only a part of the input signal that’s described in part V. Using (3), we receive the probability function p(X_r), depicted in Fig.3.

Because T₁ and T₃ are negative, d₁ and d₃ are also negative. As we can see from Fig. 3, the

	₄
Property (1) is not satisfied, but property (2) is satisfied because	åd_i = d and
	^{i = 4}

the integral from delta-function d is 1. How can we interpret this result?

The sign "plus" or "minus" for the value of probability function p(X_r) determines direct or inverse integrations for stairs of reference signal X_r. Therefore, the probability functions p(X_r) describe not only the random variable X_r, but also the composition of this random variable and the method of its transformation. Let us call this composition a “transformer process.” For some of its values, the probability function p(X_r) can be negative. The negative probability function may be useful for researching nonlinear systems. Here, the results of direct integration for a positive variable X and inverse integration for a negative variable -X do not coincide.

There are papers that focus on the use of negative probabilities, including works of Nobel prize-winners Paul Dirac and Richard Feynman. The best and last review of these papers is in [4]. But negative probabilities did not win recognition yet, partly because of contradictions with Kolmogorov probability theory. But let’s use a quotation from [4]: “There were no experiments where we can say that there were directly “detected negative probabilities”. In this research we found the interpretation of positive and negative probabilities as direct and inverse integration of signals considered as random variables.

IX. Conclusions
We consider the input x(t) and the reference X_r signals as random variables. This permits us to use the apparatus of probability theory and receive new results.

The error of measurement DV is minimized when the duration and the moments of input signal x(t) are the same as the duration and the moments for staircase reference signals X_r. When applying these conditions to the chosen values of stairs X_i, we find the durations T_i of these stairs by solving a system of equations (12).

If the reference signal X_r covers the most part of the input signal x(t), the probability function p(X_r) describes the reference X_r. We compare the reading W_r of the integral from the reference signal X_r to the reading W(t) of the integral from the input signal x(t) for the determination of x(t) or V(t).

If the reference signal X_r covers only a part of the input signal x(t), the probability function p(X_r) describes the composition of the reference signal and the method of its treatment, i.e. the transformer process. Then some values T_i are negative and we can obtain the reading W_r only by combining the direct and inverse integrations. The transformer process may be useful for further research of nonlinear systems.

Integration of the staircase reference signal can be substituted by operations of multiplication and addition (and subtraction, in case of inverse integration). Then the measurement may be continuous.

Error of measurement is lesser, nearer the moments of the real input signal to their expected values. The exact evaluation of the first moment is especially important. Error of measurement can be reduced to 10^-5 if the moments M_i are determined at time of measurement.

References
[1]. Sergey Sibeykin. Method and device of determination of an input signal which changes in time. USA, Patent US No. 6,240,435 b1, Date of Patent May 29, 2001.
[2] Ash, Robert B. Probability and Measure theory. 2000, pp.174–176.
[3] Papoulis, A. Probability, Random Variables, and Stochastic Processes.1965, pp.88–103.
[4] Espen Gaarder Haug. Derivatives Models on Models. 2007, pp. 317-333.

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